The Physics of a Foul Down the Line

It hasn’t been a well-played game, but it has been fun to watch. The lead has gone back and forth, but now in the bottom of the ninth your boys are down by a couple of runs. In keeping with the theme of this contest, they have managed to load the bases with two out. The stars seem to be aligning; your last pinch hitter is a righty to face their southpaw. He battles to a two-two count then lashes a drive down the right field line. The fans come to their feet screaming in anticipation of a bases-clearing double to win the game.

However, seemingly at the last possible second, the ball fades and lands just inches foul. The next pitch is a gutsy slider. Your guy freezes as the ump calls strike three.

The next evening, during a game of eight ball, you’re retelling the tale of the bottom of the ninth. Ironically you grumble, “Why do balls hit down the line always seem to drift foul?” just as you sink a tricky bank shot. As pool players know, the angle the ball makes with the side as it approaches is equal to the angle it leaves. Physicists call this the Law of Reflection and it is a good starting point for understanding why balls hit down the line tend to move toward foul territory.

Let’s replace the cue ball heading toward the rail with a fastball screaming toward the bat. In figure 2, you see the bat in three different positions as it contacts the ball. One will hit the ball to left field, one to center, and one to right. This explains why you want to hit the ball out in front of the plate to pull it, while you wait for the ball to get deep into the hitting zone to drive it the opposite way. Also, since the bat speeds up until it collides with the ball, a righty will generally hit the ball harder when he pulls pull it than when he goes to right field.

Next, we need to look in detail at the forces acting on the ball during the collision with the bat. Figure 3 shows the ball in contact with the bat for a ball hit to left field and for a ball hit to right field. In both cases, there is a force on the ball perpendicular to the bat and a frictional force on the ball along the bat. The perpendicular force is primarily responsible for propelling the ball toward the outfield. The frictional force creates sidespin on the ball. Notice in both cases, the spin is such that the front of the ball is moving toward foul territory.

A well-struck ball is usually hit slightly above the center of the bat. This misalignment of the bat and ball also creates spin. It is a bit hard to visualize, but stay with me. When a batter hits a ball to the opposite field, he usually has the barrel of the bat below the handle. So, the spin created by hitting the ball off the top of the bat has a substantial portion of sidespin toward foul territory as well.

Okay, even I’m having trouble seeing this.

Let’s look at an extreme case (not realistic) of a righty hitting the ball the opposite way with the bat completely vertical – that is, the barrel is as far below the handle as possible. From the top it would look like the figure 4. The pitch would have come down from the top of the page and the bat would be moving up the page. Notice sidespin is created by the frictional force. The front of the ball is spinning toward foul territory. Even if the bat is more horizontal, the ball will have some sidespin toward the right field line. If the right-handed batter pulls the ball, the resulting spin is again toward the line, but I’ll let you draw that picture for yourself.

Now we have learned that a ball hit down either line has sidespin from two sources; the angle of the bat with respect to the pitch and the ball hitting the bat above the mid-line when the barrel is below the handle. Both contributions result in spin where the front of the ball is moving toward foul territory.

When a ball is spinning through the air, it feels a Magnus force. The Magnus force causes a curveball with topspin to drop faster than gravity demands and causes a fastball thrown with backspin to drop less. It is no different for a batted ball with sidespin. It will feel a sideways Magnus force. Whether the ball is hit along the left field or right field line, the Magnus force created by the air acting on the spinning ball will cause it to veer toward foul territory.

So, any ball hit down either line will drift toward foul territory. Outfielders are well aware of this tendency. Right fielders know that balls hit down the line by right-handed batters will bend more than balls hit by left-handers. In both cases, the angle of the bat with respect to the pitch causes about the same amount of spin.

However, when a hitter goes the opposite way, the barrel of the bat is usually lower than when he pulls the ball. So the sidespin due to the ball hitting above the midline of the bat is much greater for balls hit the other way than with a pulled shot. In addition, when a batter pulls the ball it generally travels faster than when he goes the opposite way. Together then, a righty’s slap to right field will curve more toward the right field line than a ball pulled by a lefty.

In summary, we see why a ball hit the opposite way is more likely to go foul than a ball pulled down the line. So as it turns out, in baseball as well as in life, the way you see things depends upon the “spin” put on them.

Comments

If the ball hits above the center of the bat and the bat head is tilted below the hands, then the resulting sidespin for balls hit toward CF causes a slicing movement, breaking toward RF when hit by RHH (or to LF when hit by LHH). You can actually measure this effect quantitatively by combining HITf/x data with hittracker data (for home runs). I discussed this effect (and showed the data) at the 2009 PITCHf/x summit. The physics is exactly the same as Dave explains in the article.

I think a fun example to look at would be Hunter Pence’s broken-bat hit against the Cardinals in game 7 of the NLDS in 2012. The ball hit his bat 3(!) times, which caused it to spin the opposite way it should have- making Pete Kozma misread the play.

Because spin is the meat of the article, it should be mentioned than spin imparted to the cue ball, called english, can have a dramatic effect on the angle of the cue ball coming off the rail. In the diagram, if that were the cue ball, hitting the cue ball on the right of center will lengthen the angle, hitting it on the left of center will shorten the angle. Hitting the cue ball high or low will have respectively the same effect to a lesser degree. Heavy left english could potentially make the cue ball return off the rail perpendicular to the rail, and the addition of low english would curve the cue ball slightly, reducing the angle into the rail, and reducing the outgoing angle even more.

English applied to the cue ball is transferred to the object ball in reverse, though to a lesser degree as friction allows, and so the object ball will depart from the angle of reflection according to the english applied to the cue ball. This is extremely useful in maneuvering the cue ball and/or object ball around other balls that might be in the way.

Your comments on pool are interesting. They give me a chance to point out that research has shown that for a baseball colliding with a bat at game speeds, the initial spin of the ball has at most a very small affect on the spin of the ball as it leaves the bat. See Alan Nathan’s paper – http://baseball.physics.illinois.edu/ProcediaEngineering34Spin.pdf